Four point charges are placed on the x-axis: +4q at x = -3d , -2q at x = -d , +2q at x = d , and -4q at x = 3d . An external agent slowly moves a test charge Q from point A(r, 0) to point B(0, r) , where r d . The work done by the external agent is given by [ Q q d 4 ₀ r^2 ] . The value of is ______.

Step-by-step solution

The system consists of two collinear dipoles. The inner charges -2q at x = -d and +2q at x = d form a dipole of moment p₁ = (2q)(2d) = 4qd in the +x direction. The outer charges -4q at x = 3d and +4q at x = -3d form a dipole of moment p₂ = (4q)(6d) = 24qd in the -x direction. The net dipole moment is p _ net = 4qd i - 24qd i = -20qd i . The potential at point A(r, 0) on the axis of the dipole is V_A = 1 4 ₀ p _ net i r^2 = - 1 4 ₀ 20qd r^2 . The potential at point B(0, r) on the equatorial line of the dipole is V_B